We study the singularities of the isotropic skeleton of a Weinstein manifold in relation to Nadler's program of arboreal singularities. By deforming the skeleton via homotopies of the Weinstein structure, we produce a Morse-Bott* representative of the Weinstein homotopy class whose stratified skeleton determines its symplectic neighborhood. We then study the singularities of the skeleta in this class and show that after a certain type of generic perturbation either (1) these singularities fall into the class of (signed Lagrangian versions of) Nadler's arboreal singularities which are combinatorially classified into finitely many types in a given dimension or (2) there are singularities of tangency in associated front projections. We then turn to the singularities of tangency to try to reduce them also to collections of arboreal singularities. We give a general localization procedure to isolate the Liouville flow to a neighborhood of these non-arboreal singularities, and then show how to replace the simplest singularities of tangency (those of type Σ 1,0 ) by arboreal singularities.

Abstract. We give finiteness results and some classifications up to diffeomorphism of minimal strong symplectic fillings of Seifert fibered spaces over S 2 satisfying certain conditions, with a fixed natural contact structure. In some cases we can prove that all symplectic fillings are obtained by rational blow-downs of a plumbing of spheres. In other cases, we produce new manifolds with convex symplectic boundary, thus yielding new cut-and-paste operations on symplectic manifolds containing certain configurations of symplectic spheres.

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to 5, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curve arguments, together with birational transformations in the symplectic setting.

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines.The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to RP 1 ) in the real projective plane. In this paper we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres.

In this paper, we compute the Khovanov homology over Q for (p, −p, q) pretzel knots for 3 ≤ p ≤ 15, p odd, and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p, −p, q) pretzel knots. These computations reveal that these knots have thin Khovanov homology (over Q or Z). Because Greene has shown that these knots are not quasi-alternating, this provides an infinite class of non-quasi-alternating knots with thin Khovanov homology.Since α, β, and γ preserve quantum grading, basic linear algebra implies thatThis results in 4 possibilities for the isomorphism class of Kh 3 (D). Using the exact sequence (2), we can determine the isomorphism class of Kh 4 (D) corresponding to each of these four possibilities:3 (D) = Q (2−q) ⊕ Q 2 (4−q) Kh 4 (D) = Q (4−q) , Kh 3 (D) = Q (2−q) ⊕ Q (4−q) ⊕ Q (6−q) Kh 4 (D) = Q (6−q) , Kh 3 (D) = Q (2−q) ⊕ Q 2 (4−q) ⊕ Q (6−q) Kh 4 (D) = Q (4−q) ⊕ Q (6−q) .